\section*{Naïve bayes Classifier with Laplace Smoothing}

The symbols in Naïve bayes classifier with Laplace smoothing are:
\begin{itemize}
\item $N$ is the number of reviews.
\item $c$ is a sentiment.
\item $C$ is collection of sentiment classes. Example in our problem the sentiment classes are negative and positive.
\item $x_i$ is a word in the vocabulary.
\item $X$ is the vocabulary.
\item $|X|$ is size of the vocabulary.
\item $N(c)$ is the number of reviews that have the $c$.
\item $N(x_i , c)$ is the number of times the word $x_i$ is in $c$.
\item $|C|$ is the number of classes.
\end{itemize}

The function for Naïve bayes Classifier with Laplace smoothing is: $log_{10} \; s(x,c) = log_10 \; P(c) + \Sigma_(i=1)^(n) \; p(x_i | c)$

Calculate $p(c)$ for all $c \in C$ by doing: $p(c)= \frac{N(c)+1}{N+|C|}$

Calculate $p(x_i|c)$ for all possible words in vocabulary $x_i \in X$ and all possible classes $c \in C$ by doing: $p(x_i|c)= \frac{N(x_i , c)+1}{N(c)+|X|}$

